Mix the grated cucumber in a bowl with the Greek yogurt and spices. Refrigerate until ready to serve.
For the Chicken:
Preheat grill to medium-high heat. Pound chicken to about 1/3 inch thick. Sprinkle both sides with salt, pepper, and the Greek seasoning. Cook on both sides for 4-6 minutes until done. Let the meat rest for 5 minutes and then slice into thin strips.
For the Serving:
Assemble gyros by topping each pita flat bread with some chicken, cucumber, tomatoes, lettuce, red onions, feta cheese, and Tzatziki Sauce. Serve immediately.*
Notes
*For easy eating you can wrap the bottom half of each gyro in foil to help hold it together.